RBSE Solutions Class 8 Maths Chapter 9 Algebraic Expressions Exercise 9.2

Rajasthan Board RBSE Class 8 Maths Chapter 9 Algebraic Expressions Ex 9.2

Question 1. Multiply the given binomials
(i) \( (2x + 5) \) and \( (3x - 7) \)
(ii) \( (x - 8) \) and \( (3y + 5) \)
(iii) \( (1.5p - 0.5q) \) and \( (1.5p + 0.5q) \)
(iv) \( (a + 3b) \) and \( (x + 5) \)
(v) \( (2lm + 3m^2) \) and \( (3lm - 5m^2) \)
(vi) \( (\frac{3}{4}a^2 + 3b^2) \) and \( (4a^2 - \frac{5}{3}b^2) \)
Answer:
(i) To multiply \( (2x + 5) \) and \( (3x - 7) \):
\( = 2x(3x - 7) + 5(3x - 7) \)
\( = (2x \times 3x) - (2x \times 7) + (5 \times 3x) - (5 \times 7) \)
\( = 6x^2 - 14x + 15x - 35 \)
\( = 6x^2 + x - 35 \)
(ii) To multiply \( (x - 8) \) and \( (3y + 5) \):
\( = x(3y + 5) - 8(3y + 5) \)
\( = (x \times 3y) + (x \times 5) - (8 \times 3y) - (8 \times 5) \)
\( = 3xy + 5x - 24y - 40 \)
(iii) To multiply \( (1.5p - 0.5q) \) and \( (1.5p + 0.5q) \):
This uses the identity \( (A - B)(A + B) = A^2 - B^2 \).
\( = (1.5p)^2 - (0.5q)^2 \)
\( = 2.25p^2 - 0.25q^2 \)
(iv) To multiply \( (a + 3b) \) and \( (x + 5) \):
\( = a(x + 5) + 3b(x + 5) \)
\( = (a \times x) + (a \times 5) + (3b \times x) + (3b \times 5) \)
\( = ax + 5a + 3bx + 15b \)
(v) To multiply \( (2lm + 3m^2) \) and \( (3lm - 5m^2) \):
\( = 2lm(3lm - 5m^2) + 3m^2(3lm - 5m^2) \)
\( = (2lm \times 3lm) - (2lm \times 5m^2) + (3m^2 \times 3lm) - (3m^2 \times 5m^2) \)
\( = 6l^2m^2 - 10lm^3 + 9lm^3 - 15m^4 \)
\( = 6l^2m^2 - lm^3 - 15m^4 \)
(vi) To multiply \( (\frac{3}{4}a^2 + 3b^2) \) and \( (4a^2 - \frac{5}{3}b^2) \):
\( = \frac{3}{4}a^2(4a^2 - \frac{5}{3}b^2) + 3b^2(4a^2 - \frac{5}{3}b^2) \)
\( = (\frac{3}{4}a^2 \times 4a^2) - (\frac{3}{4}a^2 \times \frac{5}{3}b^2) + (3b^2 \times 4a^2) - (3b^2 \times \frac{5}{3}b^2) \)
\( = 3a^4 - \frac{5}{4}a^2b^2 + 12a^2b^2 - 5b^4 \)
\( = 3a^4 + (-\frac{5}{4} + 12)a^2b^2 - 5b^4 \)
\( = 3a^4 + (-\frac{5}{4} + \frac{48}{4})a^2b^2 - 5b^4 \)
\( = 3a^4 + \frac{43}{4}a^2b^2 - 5b^4 \)
In simple words: To multiply two binomials, we use the distributive property. This means we multiply each term in the first binomial by each term in the second binomial and then combine any similar terms. It's like opening two brackets by multiplying everything inside.

🎯 Exam Tip: Remember to apply the distributive property correctly, multiplying every term from the first bracket with every term from the second. Pay close attention to signs and exponents.

Question 2. Find the product
(i) \( (3x + 8) (5 - 2x) \)
(ii) \( (x + 3y) (3x - y) \)
(iii) \( (a^2 + b) (a + b^2) \)
(iv) \( (p^2 - q^2) (2p + q) \)
Answer:
(i) To find the product of \( (3x + 8) \) and \( (5 - 2x) \):
\( = 3x(5 - 2x) + 8(5 - 2x) \)
\( = (3x \times 5) - (3x \times 2x) + (8 \times 5) - (8 \times 2x) \)
\( = 15x - 6x^2 + 40 - 16x \)
\( = -6x^2 + (15x - 16x) + 40 \)
\( = -6x^2 - x + 40 \)
(ii) To find the product of \( (x + 3y) \) and \( (3x - y) \):
\( = x(3x - y) + 3y(3x - y) \)
\( = (x \times 3x) - (x \times y) + (3y \times 3x) - (3y \times y) \)
\( = 3x^2 - xy + 9xy - 3y^2 \)
\( = 3x^2 + (-xy + 9xy) - 3y^2 \)
\( = 3x^2 + 8xy - 3y^2 \)
(iii) To find the product of \( (a^2 + b) \) and \( (a + b^2) \):
\( = a^2(a + b^2) + b(a + b^2) \)
\( = (a^2 \times a) + (a^2 \times b^2) + (b \times a) + (b \times b^2) \)
\( = a^3 + a^2b^2 + ab + b^3 \)
(iv) To find the product of \( (p^2 - q^2) \) and \( (2p + q) \):
\( = p^2(2p + q) - q^2(2p + q) \)
\( = (p^2 \times 2p) + (p^2 \times q) - (q^2 \times 2p) - (q^2 \times q) \)
\( = 2p^3 + p^2q - 2q^2p - q^3 \)
In simple words: Finding the product of algebraic expressions means multiplying them together. We use the distribution method where each part of the first expression multiplies with each part of the second expression. Remember to combine terms that are similar after all multiplications are done.

🎯 Exam Tip: Be careful with negative signs when distributing terms. A common mistake is forgetting to distribute the negative sign from a term correctly.

Question 3. Simplify
(i) \( (x + 5) (x - 7) + 35 \)
(ii) \( (a^2 - 3) (b^2 + 3) + 5 \)
(iii) \( (t + s^2) (t^2 - s) \)
(iv) \( (a + b) (c - d) + (a - b) (c + d) + 2 (ac + bd) \)
(v) \( (a + b) (a^2 - ab + b^2) \)
(vi) \( (a + b + c) (a + b - c) \)
(vii) \( (a + b) (a - b) - a^2 + b^2 \)
Answer:
(i) To simplify \( (x + 5) (x - 7) + 35 \):
First, multiply the binomials:
\( = x(x - 7) + 5(x - 7) + 35 \)
\( = x^2 - 7x + 5x - 35 + 35 \)
Combine like terms:
\( = x^2 + (-7x + 5x) + (-35 + 35) \)
\( = x^2 - 2x \)
(ii) To simplify \( (a^2 - 3) (b^2 + 3) + 5 \):
First, multiply the binomials:
\( = a^2(b^2 + 3) - 3(b^2 + 3) + 5 \)
\( = a^2b^2 + 3a^2 - 3b^2 - 9 + 5 \)
Combine like terms:
\( = a^2b^2 + 3a^2 - 3b^2 + (-9 + 5) \)
\( = a^2b^2 + 3a^2 - 3b^2 - 4 \)
(iii) To simplify \( (t + s^2) (t^2 - s) \):
Multiply the binomials:
\( = t(t^2 - s) + s^2(t^2 - s) \)
\( = (t \times t^2) - (t \times s) + (s^2 \times t^2) - (s^2 \times s) \)
\( = t^3 - ts + s^2t^2 - s^3 \)
(iv) To simplify \( (a + b) (c - d) + (a - b) (c + d) + 2 (ac + bd) \):
Expand each product separately:
\( (a + b) (c - d) = ac - ad + bc - bd \)
\( (a - b) (c + d) = ac + ad - bc - bd \)
\( 2 (ac + bd) = 2ac + 2bd \)
Now add them together:
\( = (ac - ad + bc - bd) + (ac + ad - bc - bd) + (2ac + 2bd) \)
Group like terms:
\( = (ac + ac + 2ac) + (-ad + ad) + (bc - bc) + (-bd - bd + 2bd) \)
\( = 4ac + 0 + 0 + 0 \)
\( = 4ac \)
(v) To simplify \( (a + b) (a^2 - ab + b^2) \):
This is a standard algebraic identity for the sum of cubes.
\( = a(a^2 - ab + b^2) + b(a^2 - ab + b^2) \)
\( = (a \times a^2) - (a \times ab) + (a \times b^2) + (b \times a^2) - (b \times ab) + (b \times b^2) \)
\( = a^3 - a^2b + ab^2 + a^2b - ab^2 + b^3 \)
Combine like terms:
\( = a^3 + (-a^2b + a^2b) + (ab^2 - ab^2) + b^3 \)
\( = a^3 + 0 + 0 + b^3 \)
\( = a^3 + b^3 \)
(vi) To simplify \( (a + b + c) (a + b - c) \):
Let \( (a + b) \) be \( X \). Then the expression is \( (X + c)(X - c) \).
Using the identity \( (A + B)(A - B) = A^2 - B^2 \):
\( = (a + b)^2 - c^2 \)
Expand \( (a + b)^2 \):
\( = (a^2 + 2ab + b^2) - c^2 \)
\( = a^2 + b^2 - c^2 + 2ab \)
(vii) To simplify \( (a + b) (a - b) - a^2 + b^2 \):
First, multiply \( (a + b) (a - b) \). This uses the identity \( (A + B)(A - B) = A^2 - B^2 \).
\( = (a^2 - b^2) - a^2 + b^2 \)
Remove the parentheses:
\( = a^2 - b^2 - a^2 + b^2 \)
Combine like terms:
\( = (a^2 - a^2) + (-b^2 + b^2) \)
\( = 0 + 0 \)
\( = 0 \)
In simple words: Simplifying expressions means making them as simple as possible by doing all the multiplications and additions, and combining similar terms. Often, using algebraic rules or identities can make this process faster. If you see terms that are opposites, they will cancel each other out.

🎯 Exam Tip: Always look for algebraic identities like \( (A+B)(A-B) = A^2-B^2 \) or \( (A+B)^2 = A^2+2AB+B^2 \) before doing direct multiplication, as they can simplify calculations. Remember to keep track of all positive and negative signs.